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E. Gentle (1998), Numerical Linear Algebra for Applications in Statistics, Springer-Verlag, New York. F. A. Graybill (1961), An Introduction to Linear Statistical Models, Vol. I, McGraw–Hill, New York. F. A. , 1969. A. S. Hadi (1996), Matrix Algebra as a Tool, Wadsworth, Belmont, CA. D. A. Harville (1997), Matrix Algebra from a Statistician’s Perspective, Springer-Verlag, New York. M. J. R. , 1986. R. Horn and C. R. Johnson (1985), Matrix Analysis, Cambridge University Press, Cambridge, UK. H. O.

After making such a transformation, the SSP method is used to obtain the unbiased estimator of the square of the population mean of the transformed data, which will have the form Q(Y + θ1). Then it follows that 2 (µ + θ ) = Q(Y + θ1) = Q(Y ) + 2G(Y )θ + θ 2 , where G(Y ) is a linear function of the observations. 2) 26 Chapter 10. 2) is (µˆ − G(Y ))2 = Q(Y ) − {G(Y )}2 = µˆ 2 − {G(Y )}2 . This suggests the estimator of the population mean as µˆ = G(Y ). 3) is an unbiased estimator of µ. 2), we have µˆ = G(Y ) = [Q(Y + θ1) − Q(Y ) − θ 2 ]/2θ, so that E(µ) ˆ = [E{Q(Y + θ1)} − E{Q(Y )} − θ 2 ]/2θ = [(µ + θ )2 − µ2 − θ 2 ]/2θ = µ.

Henderson’s Method III Remarks: (i) An alternative formulation of Henderson’s Method III can be given as follows (Verdooren, 1980). 1) in the following form: Y = Xα + U1 β1 + U2 β2 + · · · + Up βp , where X is an N × q matrix of known fixed numbers, q ≤ N, Ui is an N × mi matrix of known fixed numbers, mi ≤ N, α is a q-vector of fixed effects, and βi is an mi -vector of random effects. We further assume that E(βi ) = 0, βi s are uncorrelated, and E(βi βi ) p 2 = σi2 Imi . The assumptions imply that Var(Y ) = i=1 σi Ui Ui = p 2 i=1 σi Vi where Vi = Ui Ui .

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